Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?

Suppose we are given a vector f in a class F N, e. g. , a class of digital signals or digital images. How many linear measurements do we need to make about f to be able to recover f to within precision in the Euclidean (₂) metric? This paper shows that if the objects of interest are sparse in a fixed basis or compressible, then it is possible to reconstruct f to within very high accuracy from a small number of random measurements by solving a simple linear program. More precisely, suppose that the nth largest entry of the vector f (or of its coefficients in a fixed basis) obeys f (₍) R n^-1/p, where R > 0 and p > 0. Suppose that we take measurements yₖ = f, Xₖ, k = 1, , K, where the Xₖ are N-dimensional Gaussian vectors with independent standard normal entries. Then for each f obeying the decay estimate above for some 0 and with overwhelming probability, our reconstruction f^, defined as the solution to the constraints yₖ = f^, Xₖ with minimal ₁ norm, obeys f - f^ 䃒 Cₚ R (K/ N) ^-r, r = 1/p - 1/2. There is a sense in which this result is optimal; it is generally impossible to obtain a higher accuracy from any set of K measurements whatsoever. The methodology extends to various other random measurement ensembles; for example, we show that similar results hold if one observes a few randomly sampled Fourier coefficients of f. In fact, the results are quite general and require only two hypotheses on the measurement ensemble which are detailed.